多项式exp

  1 #include <bits/stdc++.h>
  2 
  3 using namespace std;
  4 
  5 typedef long long ll;
  6 const int N = 262144,mod = 998244353,G = 3;
  7 
  8 void read(int&x){
  9     x = 0;char ch = getchar();
 10     while (ch < ‘0‘ || ‘9‘ < ch) ch = getchar();
 11     while (‘0‘ <= ch&& ch <= ‘9‘) x = x * 10 + ch - ‘0‘,ch = getchar(); 
 12 }
 13 int power(int x,int y,int z){
 14     int ans = 1;
 15     while (y){
 16         if (y&1) ans = (ll)ans*x%z;
 17         x = (ll)x*x%z;
 18         y >>= 1;
 19     }
 20     return ans;
 21 }
 22 int add(int x,int y){
 23     x += y;
 24     while (x >= mod) x -= mod;
 25     return x;
 26 }
 27 
 28 
 29 int a[N+10],b[N+10],c[N+10],d[N+10],p[N];
 30 int pg[N+10],pg_inv[N+10],inv[N+10];
 31 int n,m;
 32 void FFT(int*,int,int);
 33 void poly_exp(int*,int*,int*,int*,int);
 34 void poly_inv(int*,int*,int*,int);
 35 void poly_ln(int*,int*,int*,int);
 36 void poly_der(int*,int);
 37 void poly_mot(int*,int); 
 38 int main(){
 39     read(n);
 40     for (int i = 0;i < n;i++) read(a[n-i-1]);
 41     for (m = 1;m < n;m<<=1);
 42     for(int i = 1;i <= (m<<1);i <<= 1){
 43         pg[i] = power(G,(mod-1)/i,mod);
 44         pg_inv[i] = power(pg[i],mod-2,mod);
 45     }
 46     inv[1] = 1;
 47     for (int i = 2;i <= N;i++)inv[i] = mod-(ll)mod/i*inv[mod%i]%mod;
 48     /* 
 49     poly_inv(a,b,c,m); 
 50 //    for (int i = 0;i < m<<1;i++)cout << a[i]<<‘ ‘;cout<<endl;
 51 //    for (int i = 0;i < m<<1;i++)cout << b[i]<<‘ ‘;cout<<endl;
 52 //    cout <<(ll)a[0]*b[0]%mod<<endl;
 53     FFT(a,1,m<<1);
 54 //    for (int i = 0;i < m<<1;i++)cout <<a[i]<<‘*‘;cout<<endl;
 55     FFT(b,1,m<<1);
 56     for(int i = 0;i <m<<1;i++) a[i] =(ll)a[i]*b[i]%mod;
 57     FFT(a,-1,m<<1);
 58 //    for (int i = 0;i < m;i++)cout << a[i]<<‘ ‘;cout<<endl;
 59     bool flag = 1;
 60     for(int i = 1;i < m;i++)if (a[i] != 0) flag = 0;
 61     if(a[0] != 1) flag = 0;
 62     puts(flag ? "YES" : "NO");
 63     */ 
 64 //    /*
 65     poly_exp(a,b,c,d,m);
 66 //    for (int i = 0;i < m<<1;i++) cout << a[i]<<‘ ‘;cout << endl;
 67 //    for (int i = 0;i < m<<1;i++)cout << b[i] <<‘ ‘;cout<<endl;
 68     /*
 69 //    b[0] = 1;b[1] = 1;b[2] = inv[2];b[3] = inv[6];
 70     poly_ln(b,c,d,m);bool flag = 1;
 71 //    for (int i = 0;i < m;i++) cout << c[i]<<‘ ‘;cout<<endl;
 72     for (int i = 0;i <m;i++)if(c[i] != a[i]) flag = 0;
 73     puts(flag ? "YES" : "NO");
 74     */
 75 //    */
 76     return 0;
 77 }
 78 void FFT(int *a,int d,int deg){
 79     for (int i = 1;i < deg;i <<= 1)
 80         for (int j = 0;j < i;j++)
 81             p[i+j] = p[j]+deg/i/2;
 82     for (int i = 0;i <deg;i++) if (p[i] > i) swap(a[i],a[p[i]]);
 83     for (int n = 2;n <= deg;n <<= 1){
 84         int wn = (d == 1 ? pg[n] : pg_inv[n]);
 85         for (int i = 0;i <deg;i += n){
 86             int w = 1;
 87             for (int j = i;j < i + n/2;j++){
 88                 int x = a[j],y = (ll)a[j+n/2]*w%mod;
 89                 a[j] = add(x,y);
 90                 a[j+n/2] = add(x,mod-y);
 91                 w = (ll)w*wn%mod;
 92             }
 93         }
 94     }
 95     if (d == -1){
 96         for (int i = 0;i < deg;i++)
 97             a[i] = (ll)a[i]*inv[deg]%mod;
 98     }
 99 }
100 void poly_der(int*a,int deg){
101     for (int i = 0;i < deg;i++)
102         a[i] = (ll)a[i+1]*(i+1)%mod;
103 }
104 void poly_mot(int*a,int deg){
105     for (int i = deg;i > 0;i--)
106         a[i] = (ll)a[i-1]*inv[i]%mod;
107     a[0] = 0;
108 }
109 void poly_inv(int*a,int*b,int*c,int deg){
110     if (deg == 1){
111         if (a[0] <= N) b[0] = inv[a[0]];else 
112         b[0] = power(a[0],mod-2,mod);
113         return;
114     }
115     poly_inv(a,b,c,deg/2);
116     int deg2 = deg<<1;
117     //B = B(2-BA)
118     for (int i = 0;i < deg;i++) c[i] = a[i];
119     for (int i = deg;i < deg2;i++) c[i] = 0;
120     FFT(c,1,deg2);
121     FFT(b,1,deg2);
122     for (int i = 0;i < deg2;i++) b[i] = (ll)b[i]*add(2,mod-(ll)b[i]*c[i]%mod)%mod;
123     FFT(b,-1,deg2);
124     for (int i = deg;i < deg2;i++) b[i] = 0;
125 }
126 void poly_ln(int*a,int*b,int*c,int deg){
127     /*
128         b = lna
129         b‘ = a‘/a
130     */
131     int deg2 = deg<<1;
132     for (int i = 0;i < deg2;i++)c[i] = 0;
133     poly_inv(a,c,b,deg);
134     for(int i = 0;i <= deg;i++) b[i] = a[i];
135     for(int i = deg+1;i < deg2;i++) b[i] = 0;
136     poly_der(b,deg);
137     FFT(b,1,deg2);
138     FFT(c,1,deg2);
139     for (int i = 0;i < deg2;i++) b[i] = (ll)b[i]*c[i]%mod;
140     FFT(b,-1,deg2);
141     for (int i = deg;i < deg2;i++) b[i] = 0;
142     poly_mot(b,deg);
143     for(int i = deg;i < deg2;i++)b[i] = 0;
144 }
145 void poly_exp(int *a,int *b,int *c,int*d,int deg){
146     //B = B(1-lnB+A)
147     if(deg == 1){
148         //if a[0] = 0
149         b[0] = 1;
150         return;
151     }
152     int deg2=deg<<1;    
153     poly_exp(a,b,c,d,deg/2);
154     for (int i = deg;i < deg2;i++) b[i] = 0;
155     poly_ln(b,c,d,deg);
156 
157     for (int i = 0;i < deg;i++) c[i] = add(a[i]+(i==0),mod-c[i]);
158 //    cout <<"1-lnB+A ";for (int i = 0;i <deg2;i++) cout << c[i]<<‘ ‘;cout<<endl;    
159     for (int i = deg;i < deg2;i++) c[i] = 0;
160     FFT(c,1,deg2);
161     FFT(b,1,deg2);
162     for(int i = 0;i < deg2;i++) b[i] = (ll)b[i]*c[i]%mod;
163     FFT(b,-1,deg2);
164     for (int i = deg;i < deg2;i++) b[i] = 0;
165 //    for (int i = 0;i < deg2;i++)cout << b[i]<<‘ ‘;cout<<"deg:"<<deg<<"\n";
166 }